On the theoretical side, many inflationary models can be responsible of the formation of PBHs with broad mass function (see e.g. Refs. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]} for an incomplete list). The usual approach is to produce a large peak in the primordial power spectrum of curvature fluctuations [2]}. When the overdensities re-enter the Hubble radius, they collapse into PBHs if they are larger than a threshold \(\delta _c\)  and the resulting mass function qualitatively inherits the width and skewness of the peak in the primordial power spectrum. It is clear that, although physically viable, this approach requires some degree of fine-tuning in scale and amplitude of the peak if it is to explain observations.
